Asymptotically normal estimation in random graphs and random structures
Consider a large random structure -- a stochastic process on the line, a random graph, a random field on the grid -- and a function that depends only on a small part of the structure. Now use a family of transformations to ‘move’ the domain of the function over the structure, and average over the collected function values. It has only been clarified fairly recently, by results in ergodic theory, that there are precise conditions under which such averages have a law of large numbers: By collecting values at different locations, we can consistently estimate (conditional) expectations. I will explain under what conditions the estimates are also asymptotically normal. Several known central limit theorems for stationary random fields, graphon models of networks, etc emerge as special cases.